Permanent V. Determinant: An Exponential Lower Bound Assuming Symmetry J.M. Landsberg and Nicolas Ressayre Texas A&M University and Univ. Lyon I ITCS 2016 1 / 8
Valiant’s conjecture Theorem (Valiant) Let P be a homogeneous polynomial of degree m in M variables. Then there exists an n and n × n matrices A 0 , A 1 , . . . , A M such that P ( y 1 , . . . , y M ) = det n ( A 0 + y 1 A 1 + · · · + y M A M ) . Write P ( y ) = det n ( A ( y )). Let dc( P ) be the smallest n that works. Let Y = ( y i j ) be an m × m matrix and let perm m ( Y ) denote the permanent, a homogeneous polynomial of degree m in M = m 2 variables. Conjecture (Valiant, 1979) dc(perm m ) grows faster than any polynomial in m. 2 / 8
State of the art dc(perm 2 ) = 2 (classical) dc(perm m ) ≥ m 2 2 (Mignon-Ressayre, 2005) dc(perm m ) ≤ 2 m − 1 (Grenet 2011, explicit expressions) dc(perm 3 ) = 7 (Alper-Bogart-Velasco 2015), In particular, Grenet’s representation for perm 3 : y 3 y 3 y 3 0 0 0 0 3 2 1 y 1 1 0 0 0 0 0 1 y 1 0 1 0 0 0 0 2 y 1 perm 3 ( y ) = det 7 0 0 1 0 0 0 3 , y 2 y 2 0 0 1 0 0 2 1 y 2 y 2 0 0 0 1 0 3 1 y 2 y 2 0 0 0 0 1 3 2 is optimal. 3 / 8
Guiding principle: Optimal expressions should have interesting geometry Geometric Complexity Theory principle: perm m and det n are special because they are determined by their symmetry groups : Let G det n be the subgroup of the group of invertible linear maps C n 2 → C n 2 preserving the determinant, the symmetry group of det n . For example: B , C : n × n matrices with det( BC ) = 1, then det n ( BXC ) = det n ( X ), and det n ( X T ) = det n ( X ). These maps generate G det n . Let G perm m be the symmetry group of perm m , a subgroup of the group of invertible linear maps C m 2 → C m 2 . For example, E , F : m × m permutation matrices or diagonal matrices with determinant one, then perm m ( EYF ) = perm m ( Y ), and perm m ( Y T ) = perm m ( Y ). These generate G perm m . Let G L perm m be the subgroup of the group of invertible linear maps C m 2 → C m 2 generated by the E ’s. 4 / 8
Equivariance Proposition (L-Ressayre) Grenet’s expressions are G L perm m -equivariant, namely, given E ∈ G L perm m , there exist n × n matrices B , C such that A Grenet , m ( EY ) = BA Grenet , m ( Y ) C. For example, let t 1 E ( t ) = t 2 . t 3 Then A Grenet , m ( E ( t ) Y ) = B ( t ) A Grenet , m ( Y ) C ( t ), where t 3 t 1 t 3 t 1 t 3 and C ( t ) = B ( t ) − 1 . B ( t ) = t 1 t 3 1 1 1 5 / 8
Main results Theorem (L-Ressayre) Among G L perm m -equivariant determinantal expressions for perm m , Grenet’s size 2 m − 1 expressions are optimal and unique up to trivialities. Theorem (L-Ressayre) There exists a G perm m -equivariant determinantal expression for � 2 m − 1 ∼ 4 m . � perm m of size m Theorem (L-Ressayre) Among G perm m -equivariant determinatal expressions for perm m , the � 2 m � − 1 expressions are optimal and unique up to trivialities. size m In particular, Valiant’s conjecture holds in the restricted model of equivariant expressions. 6 / 8
Restricted model � general case? Howe-Young duality endofunctor : The involution on the space of symmetric functions (exchanging elementary symmetric functions with complete symmetric functions) extends to modules of the general linear group. Punch line: can exchange symmetry for skew-symmetry. Proof came from first proving an analogous theorem for det m (with the extra hypothesis that rank A 0 = n − 1) and then using the endofunctor to guide the proof. Same idea was used in Efremeko-L-Schenck-Weyman: (i) quadratic limit of the method of shifted partial derivatives for Valiant’s conjecture and (ii) linear strand of the minimal free resolution of the ideal generated by subpermanents. 7 / 8
More detail on the endofunctor Idea: we know a lot about the determinant. Use the endofunctor to transfer information about the determinant to the permanent. The catch: the projection operator. Illustration: Given a linear map f : C n → C n , one obtains linear maps f ∧ k : Λ k C n → Λ k C n , whose matrix entries are the size k minors of f and whose eigenvalues are the elementary symmetric functions of the eigenvalues of f . In particular the map f ∧ n : Λ n C n = C → Λ n C is multiplication by the scalar det n ( f ). One also has linear maps f ◦ k : S k C n → S k C n , whose eigenvalues are the complete symmetric functions of the eigenvalues of f . The map f ◦ k is Howe-Young dual to f ∧ k . Project S n C n to the line spanned by the square-free monomial. The image of the map induced from f ◦ n is the permanent. 8 / 8
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